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21: 18.31 Bernstein–Szegő Polynomials

… ►In consequence, $p_n(\cos \theta )$ can be given explicitly in terms of $\rho (\cos \theta )$ and sines and cosines, provided that in the first case, in the second case, and in the third case. …

22: Daniel W. Lozier

… ►He has also served several terms as an officer of the SIAM Activity Group on Orthogonal Polynomials and Special Functions. …

23: 17.14 Constant Term Identities

§17.14 Constant Term Identities

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Rogers–Ramanujan Constant Term Identities

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24: 2.11 Remainder Terms; Stokes Phenomenon

§2.11 Remainder Terms; Stokes Phenomenon

… ►First, it is impossible to bound the tail by majorizing its terms. … ►The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … ►The numerically smallest terms are the 5th and 6th. Truncation after 5 terms yields 0. …

25: 4.10 Integrals

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26: 6.20 Approximations

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§6.20(i) Approximations in Terms of Elementary Functions

… ►

Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

…

27: 29.7 Asymptotic Expansions

… ►Formulas for additional terms can be computed with the author’s Maple program LA5; see §29.22. … ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …

28: 6.16 Mathematical Applications

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6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

ⓘ

Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

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6.16.3 $$R_n(x)=\frac{1}{2}{\int }_0^x\left(\frac{1}{\sin t}-\frac{1}{t}\right)\sin \left(2nt\right)dt.$$

ⓘ

Defines:

$R_n(x)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.4 $$R_n(x)=O\left(n^{-1}\right),$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

…

29: 22.10 Maclaurin Series

… ►Initial terms are given by … ►Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). … ►Initial terms are given by … ►Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …

30: 36.15 Methods of Computation

… ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …